Angus Sinclair stood trial in 2007 accused of murdering teenagers Christine Eadie and Helen Scott. He was cleared of the charges.
The 17-year-olds were last seen leaving the World's End bar in Edinburgh in 1977.
Scotland's Lord Advocate today applied to the High Court for authority under double jeopardy legislation to set aside the acquittal of Sinclair and prosecute him again.
It is the first application to be made since the Double Jeopardy (Scotland) Act 2011 came into force last November.
The legislation sets out conditions where an accused can be retried for a crime they were previously acquitted of, with the authority of High Court judges needed for any fresh prosecution.
The Crown Office said in a statement: "The Lord Advocate has today applied to the High Court for authority under the Double Jeopardy (Scotland) Act 2011 to set aside the acquittal of Angus Sinclair and prosecute him again for the murders of Christine Eadie and Helen Scott."
Double jeopardy cases in Scotland are handled by the cold case review unit, part of the serious and organised crime division of the Crown Office.